What is the smallest positive integer that is a multiple of both 30 and 40 but not a multiple of 16?
Solution: Since 3 and 4 are relatively prime, their least common multiple is $3\cdot4=12$.  Therefore, the least common multiple of 30 and 40 is 120.  Since $\boxed{120}$ is not divisible by 16, it is the smallest common multiple of 30 and 40 which is not divisible by 16.

Note: Every common multiple of two integers is a multiple of their least common multiple.  Therefore, it is not possible that the least common multiple is divisible by 16 but some other common multiple is not.